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Oligomeric species arising during aggregation of α-synuclein are proposed to be a major source of toxicity in Parkinson’s disease, and thus a major potential drug target. However, their mechanism of formation and role in aggregation are largely unresolved. Here we first show that, at physiological pH, α-synuclein aggregates by secondary nucleation, rather than fragmentation, and that this process is enhanced by agitation. Moreover, using a combination of single molecule and bulk level techniques, we identify secondary nucleation on the surfaces of existing fibrils, rather than formation directly from monomers, as the dominant source of oligomers. Our results highlight secondary nucleation as not only the key source of oligomers, but also the main mechanism of aggregate formation, and show that these processes take place under physiologically relevant conditions.

The formation of amyloid deposits in human tissues is a defining feature of more than 50 medical disorders, including Alzheimer’s disease. Strong genetic and histological evidence links these conditions to the process of protein aggregation, yet it has remained challenging to identify a definitive connection between aggregation and pathogenicity. Using time-resolved fluorescence microscopy of individual synthetic vesicles, we show for the Aβ42 peptide implicated in Alzheimer’s disease that the disruption of lipid bilayers correlates linearly with the time course of the levels of transient oligomers generated through secondary nucleation. These findings indicate a specific role of oligomers generated through the catalytic action of fibrillar species during the protein aggregation process in driving deleterious biological function and establish a direct causative connection between amyloid formation and its pathological effects.

The spontaneous assembly of proteins into amyloid fibrils is a phenomenon central to many increasingly common and currently incurable human disorders, including Alzheimer’s and Parkinson’s diseases. Oligomeric species form transiently during this process and not only act as essential intermediates in the assembly of new filaments but also represent major pathogenic agents in these diseases. While amyloid fibrils possess a common, defining set of physicochemical features, oligomers, by contrast, appear much more diverse, and their commonalities and differences have hitherto remained largely unexplored. Here, we use the framework of chemical kinetics to investigate their dynamical properties. By fitting experimental data for several unrelated amyloidogenic systems to newly derived mechanistic models, we find that oligomers present with a remarkably wide range of kinetic and thermodynamic stabilities but that they possess two properties that are generic: they are overwhelmingly nonfibrillar, and they predominantly dissociate back to monomers rather than maturing into fibrillar species. These discoveries change our understanding of the relationship between amyloid oligomers and amyloid fibrils and have important implications for the nature of their cellular toxicity.

Oligomeric species populated during the aggregation of the Aβ42 peptide have been identified as potent cytotoxins linked to Alzheimer's disease, but the fundamental molecular pathways that control their dynamics have yet to be elucidated. By developing a general approach that combines theory, experiment and simulation, we reveal, in molecular detail, the mechanisms of Aβ42 oligomer dynamics during amyloid fibril formation. Even though all mature amyloid fibrils must originate as oligomers, we found that most Aβ42 oligomers dissociate into their monomeric precursors without forming new fibrils. Only a minority of oligomers converts into fibrillar structures. Moreover, the heterogeneous ensemble of oligomeric species interconverts on timescales comparable to those of aggregation. Our results identify fundamentally new steps that could be targeted by therapeutic interventions designed to combat protein misfolding diseases.

A general non-binary definition for on- and off-pathway intermediates is developed, enabling comparison of amyloid oligomers' contributions to fibril formation., The misfolding and aberrant aggregation of proteins into fibrillar structures is a key factor in some of the most prevalent human diseases, including diabetes and dementia. Low molecular weight oligomers are thought to be a central factor in the pathology of these diseases, as well as critical intermediates in the fibril formation process, and as such have received much recent attention. Moreover, on-pathway oligomeric intermediates are potential targets for therapeutic strategies aimed at interrupting the fibril formation process. However, a consistent framework for distinguishing on-pathway from off-pathway oligomers has hitherto been lacking and, in particular, no consensus definition of on- and off-pathway oligomers is available. In this paper, we argue that a non-binary definition of oligomers' contribution to fibril-forming pathways may be more informative and we suggest a quantitative framework, in which each oligomeric species is assigned a value between 0 and 1 describing its relative contribution to the formation of fibrils. First, we clarify the distinction between oligomers and fibrils, and then we use the formalism of reaction networks to develop a general definition for on-pathway oligomers, that yields meaningful classifications in the context of amyloid formation. By applying these concepts to Monte Carlo simulations of a minimal aggregating system, and by revisiting several previous studies of amyloid oligomers in light of our new framework, we demonstrate how to perform these classifications in practice. For each oligomeric species we obtain the degree to which it is on-pathway, highlighting the most effective pharmaceutical targets for the inhibition of amyloid fibril formation.

The self-assembly of peptides and proteins into amyloid fibrils plays a causative role in a wide range of increasingly common and currently incurable diseases. The molecular mechanisms underlying this process have recently been discovered, prompting the development of drugs that inhibit specific reaction steps as possible treatments for some of these disorders. A crucial part of treatment design is to determine how much drug to give and when to give it, informed by its efficacy and intrinsic toxicity. Since amyloid formation does not proceed at the same pace in different individuals, it is also important that treatment design is informed by local measurements of the extent of protein aggregation. Here, we use stochastic optimal control theory to determine treatment regimens for inhibitory drugs targeting several key reaction steps in protein aggregation, explicitly taking into account variability in the reaction kinetics. We demonstrate how these regimens may be updated "on the fly" as new measurements of the protein aggregate concentration become available, in principle, enabling treatments to be tailored to the individual. We find that treatment timing, duration, and drug dosage all depend strongly on the particular reaction step being targeted. Moreover, for some kinds of inhibitory drugs, the optimal regimen exhibits high sensitivity to stochastic fluctuations. Feedback controls tailored to the individual may therefore substantially increase the effectiveness of future treatments.

Funder: Lindemann Trust Fellowship, English-Speaking Union, Knowledge of the mechanisms of assembly of amyloid proteins into aggregates is of central importance in building an understanding of neurodegenerative disease. Given that oligomeric intermediates formed during the aggregation reaction are believed to be the major toxic species, methods to track such intermediates are clearly needed. Here we present a method, electron paramagnetic resonance (EPR), by which the amount of intermediates can be measured over the course of the aggregation, directly in the reacting solution, without the need for separation. We use this approach to investigate the aggregation of α-synuclein (αS), a synaptic protein implicated in Parkinson’s disease and find a large population of oligomeric species. Our results show that these are primary oligomers, formed directly from monomeric species, rather than oligomers formed by secondary nucleation processes, and that they are short-lived, the majority of them dissociates rather than converts to fibrils. As demonstrated here, EPR offers the means to detect such short-lived intermediate species directly in situ. As it relies only on the change in size of the detected species, it will be applicable to a wide range of self-assembling systems, making accessible the kinetics of intermediates and thus allowing the determination of their rates of formation and conversion, key processes in the self-assembly reaction.

Protein self-assembly into amyloid fibrils underlies several neurodegenerative conditions, including Alzheimer’s and Parkinson’s diseases. It has become apparent that the small oligomers formed during this process constitute neurotoxic molecular species associated with amyloid aggregation. Targeting the formation of oligomers represents, therefore, a possible therapeutic avenue to combat these diseases. However, it remains challenging to establish which microscopic steps should be targeted to suppress most effectively the generation of oligomeric aggregates. Recently, we have developed a kinetic model of oligomer dynamics during amyloid aggregation. Here, we use this approach to derive explicit scaling relationships that reveal how key features of the time evolution of oligomers, including oligomer peak concentration and lifetime, are controlled by the different rate parameters. We discuss the therapeutic implications of our framework by predicting changes in oligomer concentrations when the rates of the individual microscopic events are varied. Our results identify the kinetic parameters that control most effectively the generation of oligomers, thus opening a new path for the systematic rational design of therapeutic strategies against amyloid-related diseases.

Self-assembling peptide-based hydrogels are a class of tunable soft materials that have been shown to be highly useful for a number of biomedical applications. The dynamic formation of the supramolecular fibrils that compose these materials has heretofore remained poorly characterized. A better understanding of this process would provide important insights into the behavior of these systems and could aid in the rational design of new peptide hydrogels. Here, we report the determination of the microscopic steps that underpin the self-assembly of a hydrogel-forming peptide, SgI37-49. Using theoretical models of linear polymerization to analyze the kinetic self-assembly data, we show that SgI37-49 fibril formation is driven by fibril-catalyzed secondary nucleation and that all the microscopic processes involved in SgI37-49 self-assembly display an enzyme-like saturation behavior. Moreover, this analysis allows us to quantify the rates of the underlying processes at different peptide concentrations and to calculate the time evolution of these reaction rates over the time course of self-assembly. We demonstrate here a new mechanistic approach for the study of self-assembling hydrogel-forming peptides, which is complementary to commonly used materials science characterization techniques.

The formation of amyloid fibrils from soluble peptide is a hallmark of many neurodegenerative diseases such as Alzheimer's and Parkinson's diseases. Characterization of the microscopic reaction processes that underlie these phenomena have yielded insights into the progression of such diseases and may inform rational approaches for the design of drugs to halt them. Experimental evidence suggests that most of these reaction processes are intrinsically catalytic in nature and may display enzymelike saturation effects under conditions typical of biological systems, yet a unified modeling framework accounting for these saturation effects is still lacking. In this paper, we therefore present a universal kinetic model for biofilament formation in which every fundamental process in the reaction network can be catalytic. The single closed-form expression derived is capable of describing with high accuracy a wide range of mechanisms of biofilament formation and providing the first integrated rate law of a system in which multiple reaction processes are saturated. Moreover, its unprecedented mathematical simplicity permits us to very clearly interpret the effects of increasing saturation on the overall kinetics. The effectiveness of the model is illustrated by fitting it to the data of in vitro Aβ40 aggregation. Remarkably, we find that primary nucleation becomes saturated, demonstrating that it must be heterogeneous, occurring at interfaces and not in solution.

Oligomeric aggregates populated during the aggregation of the Aβ42 peptide have been identified as potent cytotoxins linked to Alzheimer’s disease, but the fundamental molecular pathways that control their dynamics have yet to be elucidated. By developing a general approach combining theory, experiment, and simulation, we reveal in molecular detail the mechanisms of Aβ42 oligomer dynamics during amyloid fibril formation. Even though all mature amyloid fibrils must originate as oligomers, we find that most Aβ42 oligomers dissociate to their monomeric precursors without forming new fibrils. Only a minority of oligomers converts into fibrillar species. Moreover, the heterogeneous ensemble of oligomeric species interconverts on timescales comparable to aggregation. Our results identify fundamentally new steps that could be targeted by therapeutic interventions designed to combat protein misfolding diseases.

Small oligomers of the protein α-synuclein (αS) are highly cytotoxic species associated with Parkinson's disease (PD). In addition, αS can form co-aggregates with its mutational variants and with other proteins such as amyloid-β (Aβ) and tau, which are implicated in Alzheimer's disease. The processes of self-oligomerization and co-oligomerization of αS are, however, challenging to study quantitatively. Here, we have utilized single-molecule techniques to measure the equilibrium populations of oligomers formed in vitro by mixtures of wild-type αS with its mutational variants and with Aβ40, Aβ42, and a fragment of tau. Using a statistical mechanical model, we find that co-oligomer formation is generally more favorable than self-oligomer formation at equilibrium. Furthermore, self-oligomers more potently disrupt lipid membranes than do co-oligomers. However, this difference is sometimes outweighed by the greater formation propensity of co-oligomers when multiple proteins coexist. Our results suggest that co-oligomer formation may be important in PD and related neurodegenerative diseases.

Localized deformation patterns are a common motif in morphogenesis and are increasingly finding applications in materials science and engineering, in such instances as mechanical memories. Here, we describe the emergence of spatially localized deformations in a minimal mechanical system by exploring the impact of growth and shear on the conformation of a semi-flexible filament connected to a pliable shearable substrate. We combine numerical simulations of a discrete rod model with theoretical analysis of the differential equations recovered in the continuum limit to quantify (in the form of scaling laws) how geometry, mechanics and growth act together to give rise to such localized structures in this system. We find that spatially localized deformations along the filament emerge for intermediate shear modulus and increasing growth. Finally, we use experiments on a 3D-printed multi-material model system to demonstrate that external control of the amount of shear and growth may be used to regulate the spatial extent of the localized strain texture.

The formation of amyloid deposits in human tissues is a defining feature of more than 50 medical disorders, including Alzheimer's disease. Strong genetic and histological evidence links these conditions to the process of protein aggregation, yet it has remained challenging to identify a definitive connection between aggregation and pathogenicity. Using time-resolved fluorescence microscopy of individual synthetic vesicles, we show for the Aβ42 peptide implicated in Alzheimer's disease that the disruption of lipid bilayers correlates linearly with the time course of the levels of transient oligomers generated through secondary nucleation. These findings indicate a specific role of oligomers generated through the catalytic action of fibrillar species during the protein aggregation process in driving deleterious biological function and establish a direct causative connection between amyloid formation and its pathological effects.

We use perturbative renormalization group theory to study the kinetics of protein aggregation phenomena in a unified manner across multiple timescales. Using this approach, we find that, irrespective of the specific molecular details or experimental conditions, filamentous assembly systems display universal behavior in time. Moreover, we show that the universality classes for protein aggregation correspond to simple autocatalytic processes and that the diversity of behavior in these systems is determined solely by the reaction order for secondary nucleation with respect to the protein concentration, which labels all possible universality classes. We validate these predictions on experimental data for the aggregation of several different proteins at several different initial concentrations, which by appropriate coordinate transformations we are able to collapse onto universal kinetic growth curves. These results establish the power of the perturbative renormalization group in distilling the ultimately simple temporal behavior of complex protein aggregation systems, creating the possibility to study the kinetics of general self-assembly phenomena in a unified fashion.

The aggregation of the prion protein (PrP) plays a key role in the development of prion diseases., The aggregation of the prion protein (PrP) plays a key role in the development of prion diseases. In the past decade, a similar process has been associated with other proteins, such as Aβ, tau, and α-synuclein, which participate in other neurodegenerative diseases. It is increasingly recognized that the small oligomeric species of aggregates can play an important role in the development of prion diseases. However, determining the nature of the oligomers formed during the aggregation process has been experimentally difficult due to the lack of suitable methods capable of the detection and characterization of the low level of oligomers that may form. To address this problem, we have utilized single-aggregate methods to study the early events associated with aggregation of recombinant murine PrP in vitro to approach the bona fide process in vivo. PrP aggregation resulted in the formation of thioflavin T (ThT)-inactive and ThT-active species of oligomers. The ThT-active oligomers undergo conversion from a Proteinase K (PK)-sensitive to PK-resistant conformer, from which mature fibrils can eventually emerge. Overall, our results show that single-aggregate methods can provide structural and mechanistic insights into PrP aggregation, identify the potential species that mediates cytotoxicity, and reveal that a range of distinct oligomeric species with different properties is formed during prion protein aggregation.

The formation of elongated supra-molecular structures from protein building blocks generates functional intracellular filaments, but this process is also at the heart of many neurodegenerative conditions including Alzheimer’s and Parkinson’s diseases, where it occurs in an uncontrolled manner. When observed at appropriate concentration and time scales, the chemical kinetics of filamentous protein self-assembly exhibits the remarkable property of self-similarity: the dynamics appears similar as the observation scale changes. We discuss here how this property leads to crucial simplifications of the fundamental laws governing protein filament formation and the emergence of scaling laws that provide the basis for connecting microscopic events with macroscopic realisations of such processes. In particular, we review recent developments in the modelling of linear protein self-assembly phenomena in the light of the concepts of dimensional analysis and physical self-similarity. We show how these tools and concept...

The misfolding and aggregation of proteins into linear fibrils is widespread in human biology, for example, in connection with amyloid formation and the pathology of neurodegenerative disorders such as Alzheimer's and Parkinson's diseases. The oligomeric species that are formed in the early stages of protein aggregation are of great interest, having been linked with the cellular toxicity associated with these conditions. However, these species are not characterized in any detail experimentally, and their properties are not well understood. Many of these species have been found to have approximately spherical morphology and to be held together by hydrophobic interactions. We present here an analytical statistical mechanical model of globular oligomer formation from simple idealized amphiphilic protein monomers and show that this correlates well with Monte Carlo simulations of oligomer formation. We identify the controlling parameters of the model, which are closely related to simple quantities that may be fitted directly from experiment. We predict that globular oligomers are unlikely to form at equilibrium in many polypeptide systems but instead form transiently in the early stages of amyloid formation. We contrast the globular model of oligomer formation to a well-established model of linear oligomer formation, highlighting how the differing ensemble properties of linear and globular oligomers offer a potential strategy for characterizing oligomers from experimental measurements.

Oligomeric species populated during the aggregation of the Aβ42 peptide have been identified as potent cytotoxins linked to Alzheimer's disease, but the fundamental molecular pathways that control their dynamics have yet to be elucidated. By developing a general approach that combines theory, experiment and simulation, we reveal, in molecular detail, the mechanisms of Aβ42 oligomer dynamics during amyloid fibril formation. Even though all mature amyloid fibrils must originate as oligomers, we found that most Aβ42 oligomers dissociate into their monomeric precursors without forming new fibrils. Only a minority of oligomers converts into fibrillar structures. Moreover, the heterogeneous ensemble of oligomeric species interconverts on timescales comparable to those of aggregation. Our results identify fundamentally new steps that could be targeted by therapeutic interventions designed to combat protein misfolding diseases.

The molecular mechanism of protein aggregation is of both fundamental and clinical importance as amyloid aggregates are linked to a number of neurodegenerative disorders. Such protein aggregates include macroscopic insoluble fibrils as well as small soluble oligomeric species. Time-dependent resolution of these species is prerequisite for a detailed quantitative understanding of protein aggregation; this remains challenging due to the lack of methods for detecting and characterizing transient and heterogeneous protein oligomers. Here we have used single molecule fluorescence techniques combined with mechanistic modeling to study the heparin-induced aggregation of the repeat region of tau, which forms the core region of neurofibrillary tangles found in Alzheimer’s disease. We distinguish several subpopulations of oligomers with different stability and follow their evolution during aggregation reactions as a function of temperature and concentration. Employment of techniques from chemical kinetics reveals that the two largest populations are structurally distinct from fibrils and are both kinetically and thermodynamically unstable. The first population is in rapid exchange with monomers and held together by electrostatic interactions; the second is kinetically more stable, dominates at later times, and is probably off-pathway to fibril formation. These more stable oligomers may contribute to other oligomer induced effects in the cellular environment, for example, by overloading protein quality control systems. We also show that the shortest growing filaments remain suspended in aqueous buffer and thus comprise a third, smaller population of transient oligomers with cross-β structure. Overall our data show that a diverse population of oligomers of different structures and half-lives are formed during the aggregation reaction with the great majority of oligomers formed not going on to form fibrils.

The self-assembly of polypeptides into amyloid structures is associated with a range of increasingly prevalent neurodegenerative diseases as well as with a select set of functional processes in biology. The phenomenon of self-assembly results in species with dramatically different sizes, from small oligomers to large fibrils; however, the kinetic relationship between these species is challenging to characterize. In the case of prion aggregates, these structures can self-replicate and act as infectious agents. Here we use single molecule spectroscopy to obtain quantitative information on the oligomer populations formed during aggregation of the yeast prion protein Ure2. Global analysis of the aggregation kinetics reveals the molecular mechanism underlying oligomer formation and depletion. Quantitative characterization indicates that the majority of Ure2 oligomers are relatively short-lived, and their rate of dissociation is much higher than their rate of conversion into growing fibrils. We identify an initial metastable oligomer, which can subsequently convert into a structurally distinct oligomer, which in turn converts into growing fibrils. We also show that fragmentation is responsible for the autocatalytic self-replication of Ure2 fibrils, but that preformed fibrils do not promote oligomer formation, indicating that secondary nucleation of the type observed for peptides and proteins associated with neurodegenerative disease does not occur at a significant rate for Ure2. These results establish a framework for elucidating the temporal and causal relationship between oligomers and larger fibrillar species in amyloid forming systems, and provide insights into why functional amyloid systems are not toxic to their host organisms.

Synuclein oligomers can be toxic to cells and may be responsible for cell death in Parkinson's disease. Their typically low abundance and highly heterogeneous nature, however, make such species challenging to study using traditional biochemical techniques. By combining fast-flow microfluidics with single-molecule fluorescence, we are able to rapidly follow the process by which oligomers of ?S are formed and to characterize the species themselves. We have used the technique to show that populations of oligomers with different FRET efficiencies have varying stabilities when diluted into low ionic strength solutions. Interestingly, we have found that oligomers formed early in the aggregation pathway have electrostatic repulsions that are shielded in the high ionic strength buffer and therefore dissociate when diluted into lower ionic strength solutions. This property can be used to isolate different structural groups of ?S oligomers and can help to rationalize some aspects of ?S amyloid fibril formation. (Figure Presented).

Silk fibroin is a natural protein obtained from the Bombyx mori silkworm. In addition to being the key structural component in silkworm cocoons, it also has the propensity to self-assemble in vitro into hierarchical structures with desirable properties such as high levels of mechanical strength and robustness. Furthermore, it is an appealing biopolymer due to its biocompatability, low immunogenicity, and lack of toxicity, making it a prime candidate for biomedical material applications. Here, it is demonstrated that nanofibrils formed by reconstituted silk fibroin can be engineered into supramolecular microgels using a soft lithography-based microfluidic approach. Building on these results, a potential application for these protein microgels to encapsulate and release small molecules in a controlled manner is illustrated. Taken together, these results suggest that the tailored self-assembly of biocompatible and biodegradable silk nanofibrils can be used to generate functional micromaterials for a range of potential applications in the biomedical and pharmaceutical fields.

The self-assembly of molecular building blocks into linear filaments is a common form of self-organization in nature and underlies the formation of supra-molecular polymers in a variety of contexts, including in both functional and aberrant biology. To date, attention has focused mainly on homomolecular assembly phenomena; however, it has recently become apparent that heteromolecular assemblies can be common, and, for instance, pathological protein filaments such as amyloid aggregates form in vivo in environments supporting copolymerization. Here, we present a general kinetic scheme for heteromolecular filament formation and derive closed-form analytical expressions that describe the dynamics of such systems. Our results reveal the existence of a demixing transition time controlled by the relative rates of depletion of the different aggregating species, after which predominantly homomolecular polymers are formed even when the initial solution is heteromolecular. Furthermore, these results may be applied to the analysis of experimental kinetic data on the aggregation of mixtures of proteins, to determine which fundamental reaction steps occur between unlike proteins, and to provide accurate estimates of their rate constants.

Biological systems are characterized by compartmentalization from the subcellular to the tissue level, and thus reactions in small volumes are ubiquitous in living systems. Under such conditions, statistical number fluctuations, which are commonly negligible in bulk reactions, can become dominant and lead to stochastic behavior. We present here a stochastic model of protein filament formation in small volumes. We show that two principal regimes emerge for the system behavior, a small fluctuation regime close to bulk behavior and a large fluctuation regime characterized by single rare events. Our analysis shows that in both regimes the reaction lag-time scales inversely with the system volume, unlike in bulk. Finally, we use our stochastic model to connect data from small-volume microdroplet experiments of amyloid formation to bulk aggregation rates, and show that digital analysis of an ensemble of protein aggregation reactions taking place under microconfinement provides an accurate measure of the rate of primary nucleation of protein aggregates, a process that has been challenging to quantify from conventional bulk experiments., We are grateful to St. John’s College, Cambridge (T. C. T. M., J. B. K.), the Schiff Foundation (A. J. D.), the EPSRC (K. L. S.), NSF Grant No. DMR-1310266 (D. A. W.), the Harvard MRSEC Grant No. DMR1420570 (D. A. W.), BBSRC (T. P. J. K.), ERC (T. C. T. M., T. P. J. K.), and Frances and Augustus Newman Foundation (T. P. J. K.) for financial support.

The growth of filamentous aggregates from precursor proteins is a process of central importance to both normal and aberrant biology, for instance as the driver of devastating human disorders such as Alzheimer's and Parkinson's diseases. The conventional theoretical framework for describing this class of phenomena in bulk is based upon the mean-field limit of the law of mass action, which implicitly assumes deterministic dynamics. However, protein filament formation processes under spatial confinement, such as in microdroplets or in the cellular environment, show intrinsic variability due to the molecular noise associated with small-volume effects. To account for this effect, in this paper we introduce a stochastic differential equation approach for investigating protein filament formation processes under spatial confinement. Using this framework, we study the statistical properties of stochastic aggregation curves, as well as the distribution of reaction lag-times. Moreover, we establish the gradual breakdown of the correlation between lag-time and normalized growth rate under spatial confinement. Our results establish the key role of spatial confinement in determining the onset of stochasticity in protein filament formation and offer a formalism for studying protein aggregation kinetics in small volumes in terms of the kinetic parameters describing the aggregation dynamics in bulk.

The lattice gas model of condensation in a heterogeneous pore system, represented by a random graph of cells, is studied using an exact analytical solution. A binary mixture of pore cells with different coordination numbers is shown to exhibit two phase transitions as a function of chemical potential in a certain temperature range. Heterogeneity in interaction strengths is demonstrated to reduce the critical temperature and, for large enough degree of disorder, divides the cells into ones which are either on average occupied or unoccupied. Despite treating the pore space loops in a simplified manner, the random-graph model provides a good description of condensation in porous structures containing loops. This is illustrated by considering capillary condensation in a structural model of mesoporous silica SBA-15., 22 pages, 16 figures